2.1 Symmetric extensions

As we remarked, a generic extension of a model of $\mathsf {ZFC}$ is again a model of $\mathsf {ZFC}$ . Symmetric extensions are intermediate models to generic extensions where the axiom of choice may fail.

Let ${\mathbb {P}}$ be a forcing, and let $\pi $ be an automorphism of ${\mathbb {P}}$ . The action of $\pi $ extends to the ${\mathbb {P}}$ -names by recursion:

$$\begin{align*}\pi\dot x=\{\langle\pi p,\pi\dot y\rangle\mid\langle p,\dot y\rangle\in\dot x\}.\end{align*}$$

Lemma (The Symmetry Lemma).

Let $\pi $ be an automorphism of a forcing ${\mathbb {P}}$ , and let $\dot x$ be a ${\mathbb {P}}$ -name. For every condition p,

$$\begin{align*}p\mathrel{\Vdash}\varphi(\dot x)\iff\pi p\mathrel{\Vdash}\varphi(\pi\dot x).\end{align*}$$

Fix a group ${\mathscr {G}}\leq \operatorname {\mathrm {Aut}}({\mathbb {P}})$ . We say that ${\mathscr {F}}$ is a filter of subgroups on ${\mathscr {G}}$ if it is a filter on the lattice of subgroups, namely, it is a non-empty collection of subgroups which is closed under finite intersections and supergroups. We will, unless stated otherwise, assume that it is a proper filter, i.e., the trivial group is not in ${\mathscr {F}}$ .Footnote ⁴ Finally, ${\mathscr {F}}$ is normal if whenever $\pi \in {\mathscr {G}}$ and $H\in {\mathscr {F}}$ , then $\pi H\pi ^{-1}\in {\mathscr {F}}$ as well. In most cases we are interested not necessarily in a filter, but in a filter base, and we will ignore the distinction between the two.

Call $\langle {\mathbb {P}},{\mathscr {G}},{\mathscr {F}}\rangle $ a symmetric system if ${\mathbb {P}}$ is a notion of forcing, ${\mathscr {G}}$ is a group of automorphisms of ${\mathbb {P}}$ , and ${\mathscr {F}}$ is a normal filter of subgroups on ${\mathscr {G}}$ . We shall fix a symmetric system $\langle {\mathbb {P}},{\mathscr {G}},{\mathscr {F}}\rangle $ for the rest of this section.

For a ${\mathbb {P}}$ -name, $\dot x$ , let $\operatorname {\mathrm {sym}}_{\mathscr {G}}(\dot x)$ denote the group $\{\pi \in {\mathscr {G}}\mid \pi \dot x=\dot x\}$ . If it is the case that $\operatorname {\mathrm {sym}}_{\mathscr {G}}(\dot x)\in {\mathscr {F}}$ , then we say that $\dot x$ is ${\mathscr {F}}$ -symmetric. And similarly, we say that $\dot x$ is hereditarily ${\mathscr {F}}$ -symmetric if being ${\mathscr {F}}$ -symmetric is hereditarily true for all names hereditarily appearing in $\{\dot x\}^\bullet $ .

We denote by ${\mathsf {HS}}_{\mathscr {F}}$ the class of hereditarily ${\mathscr {F}}$ -symmetric names. We denote by $\mathrel {\Vdash }^{{\mathsf {HS}}}$ the relativisation of the forcing relation to ${\mathsf {HS}}$ : we restrict the quantifiers and free variables to this class. It is not hard to check that the Symmetry Lemma applies for $\mathrel {\Vdash }^{{\mathsf {HS}}}$ , provided that we use automorphisms from ${\mathscr {G}}$ . The following theorem is a condensation of [Reference Jech6, pp. 253–254].

The class M is also called a symmetric extension of V. It turns out, as shown by Usuba [Reference Usuba22], that M is a symmetric extension of V if and only if $M=V(x)$ for some $x\in V[G]$ . We will discuss this in more detail in Section 7.

We will omit ${\mathscr {G}}$ and ${\mathscr {F}}$ from the notation and terminology when they are clear from context, which is usually what is going to happen.

2.2 Example

Let ${\mathbb {P}}$ be $\operatorname {\mathrm {Add}}(\omega ,\omega _1)$ . Namely, $p\in {\mathbb {P}}$ is a finite partial function from $\omega _1\times \omega \to 2$ . We let our ${\mathscr {G}}$ be the group of permutations of $\omega _1$ acting on ${\mathbb {P}}$ in the natural way: $\pi p(\pi \alpha ,n)=p(\alpha ,n)$ . Finally, for $E\subseteq \omega _1$ , let $\operatorname {\mathrm {fix}}(E)$ denote $\{\pi \in {\mathscr {G}}\mid \pi \restriction E=\operatorname {\mathrm {id}}\}$ , and set ${\mathscr {F}}=\{\operatorname {\mathrm {fix}}(E)\mid E\in [\omega _1]^{<\omega _1}\}$ .Footnote ⁵

For every $\alpha <\omega _1$ , let $\dot a_\alpha $ be the name of the $\alpha $ th Cohen real, that is $\dot{a}_\alpha = \{\langle p,\check n\rangle \mid p(\alpha ,n)=1\}$ , and set $\dot A=\{\dot a_\alpha \mid \alpha <\omega _1\}^\bullet $ .

As an immediate corollary, $\dot a_\alpha \in {\mathsf {HS}}$ for each $\alpha <\omega _1$ , as witnessed by $\operatorname {\mathrm {fix}}(\{\alpha \})$ , and so $\dot A\in {\mathsf {HS}}$ as well.

Proof Let $\dot f\in {\mathsf {HS}}$ , and suppose that p is a condition such that for some ordinal $\eta $ $p\mathrel {\Vdash }^{{\mathsf {HS}}}\dot f\colon \dot A\to \check \eta $ . Let E be a countable set such that $\operatorname {\mathrm {fix}}(E)\subseteq \operatorname {\mathrm {sym}}(\dot f)$ . We may also assume that $\pi \in \operatorname {\mathrm {fix}}(E)$ satisfies $\pi p=p$ by adding a finite set to E, and replacing it with $E\cup \{\alpha \mid \exists n\, \langle \alpha ,n\rangle \in \operatorname {\mathrm {dom}} p\}$ .

Fix $\alpha \notin E$ , the set $X=\{\xi <\eta \mid \exists q\leq p: q\mathrel {\Vdash }^{{\mathsf {HS}}}\dot f(\dot a_\alpha )=\check \xi \}$ is countable, as ${\mathbb {P}}$ is a c.c.c. forcing. Note that $p\mathrel {\Vdash }^{{\mathsf {HS}}}\dot f(\dot a_\alpha )\in \check X$ .

For any $\beta <\omega _1$ , let $\pi _\beta $ denote the $2$ -cycle $(\alpha \ \beta )$ . Therefore,

$$\begin{align*}\pi_\beta p\mathrel{\Vdash}^{{\mathsf{HS}}}\pi_\beta\dot f(\pi_\beta\dot a_\alpha)\in\pi_\beta\check X.\end{align*}$$

Easily, $\pi _\beta \in \operatorname {\mathrm {fix}}(E)$ if and only if $\beta \notin E$ . So for all $\beta \notin E$ , $p\mathrel {\Vdash }^{{\mathsf {HS}}}\dot f(\dot a_\beta )\in \check X$ . In particular, p must force that $\dot f$ has a countable range. As p and $\dot f$ were arbitrary, we in fact have shown that every ordinal-valued function in the symmetric extension defined on A must have a countable range.

To finish the proof we appeal to the c.c.c. of the Cohen forcing again, noting that $\omega _1$ is not collapsed, and therefore . Therefore there is no injection from A into the ordinals, as wanted.

The standard arguments are usually presented in a slightly different way. We usually extend p to a condition q, which decides the value of $\dot f(\dot a_\alpha )$ , and then show that we find $\pi \in \operatorname {\mathrm {fix}}(E)$ such that $\pi q$ is compatible with q, and $\pi \alpha \neq \alpha $ . This then shows that no extension of p can force $\dot f$ to be injective. Chain condition-based arguments are not very common in results of this type, and we hope that this article will help to popularise the idea.Footnote ⁶

2.3 Iterations of symmetric extensions

Iterating symmetric extensions is not an easy task. While some ad-hoc constructions can be found in the literature from the very early 1970s,Footnote ⁷ the only systematic development of such a framework was done by the author in [Reference Karagila9], and so far only for finite support iterations. The goal of this section is not to fully develop and explain this technique, but instead provide an intuition as to how the technique works, and what are the difficulties that need to be overcome in the case of the Bristol model.

The intuition behind iterations of symmetric extensions is as naive and simple as it can get. We want to have an iteration of forcing notions, in this case with finite support, and we want to identify a class of names which correspond to the intermediate model that we would get if we were to iterate symmetric extensions one step at a time.

Looking at a two-step iteration, say ${\mathbb {P}}{\ast \dot {\mathbb {Q}}}$ , we also need to associate ${\mathscr {G}}$ and ${\mathscr {F}}$ to ${\mathbb {P}}$ , and ${\mathbb {P}}$ -names for a group of automorphisms, ${\dot {\mathscr {H}}}{\leq }{\dot {\mathrm {Aut}}}({\dot {\mathbb {Q}}})$ , and a filter of subgroups ${\dot {\mathscr {K}}}$ on ${\dot {\mathscr {H}}}$ . Now, a ${\mathbb {P}}{\ast \dot {\mathbb {Q}}}$ -name is going to be in the iterated symmetric extension if its projection to ${\mathbb {P}}$ is in ${\mathsf {HS}}_{{\mathscr {F}}}$ ,Footnote ⁸ and it is forced by to be in ${\mathsf {HS}}_{{\mathscr {K}}}$ , that is, a hereditarily symmetric name in the second step.

But we can weaken this slightly, and much like we only require that the projected ${\mathbb {P}}$ -name is forced to be a name in ${\mathsf {HS}}_{\mathscr {K}}$ , we can similarly require that the projected name is forced to be equal to some name in ${\mathsf {HS}}_{\mathscr {F}}$ . That is, we are allowed to “mix” names from ${\mathsf {HS}}_{\mathscr {F}}$ over an antichain.Footnote ⁹ So the projected name need only be “generically equal” to an ${\mathscr {F}}$ -symmetric ${\mathbb {P}}$ -name, which itself needs only be “generically equal” to a ${\dot {\mathscr {K}}}$ -symmetric ${\dot {\mathbb {Q}}}$ -name.

Consider, for example, Cohen’s first model, this is a model similar to the example above, replacing $\omega _1$ by $\omega $ . Namely, we add an $\omega $ -sequence of Cohen reals, permute them, and consider finite stabilisers. Let $\dot a_n$ be the name for the nth real, then we can define the name

$$\begin{align*}\dot a=\{\langle p,\check m\rangle\mid\exists n(p(n,0)=1\land\forall k<n, p(k,0)=0\land p(n,m)=1)\}.\end{align*}$$

In clearer terms, this is the name for the first $\dot a_n$ which contains $0$ . We are guaranteed that $\dot a$ will be interpreted as one of the $\dot a_n$ . But it is not hard to see that $\dot a$ , as defined above, is not stabilised by fixing any finite subset of  $\omega $ .Footnote ¹⁰

Considering names like $\dot a$ seems like an unnecessary complication. But upon a closer examination of the general construction of iterations, we see that this idea is somehow necessary. Indeed, the conditions of the iteration are usually defined to be $\langle p,\dot q\rangle $ such that $p\in {\mathbb {P}}$ and with $\dot q$ coming from a “sufficiently rich” set of names. We will call iterations defined this way “Jech(-style) iterations”. Kunen, in his book, first introduces iterations more naively: the conditions are pairs $\langle p,\dot q\rangle $ such that $p\in {\mathbb {P}}$ and $p\mathrel {\Vdash }\dot q\in \dot {\mathbb {Q}}$ with $\dot q$ appearing in $\dot {\mathbb {Q}}$ . Unlike Jech iterations, this definition does not generalise well, but it is useful for understanding finite support iterations; we will refer to this as “Kunen(-style) iterations”.

When working with Jech iterations, we can quickly see that even for the conditions of ${\mathbb {P}}\ast \dot {\mathbb {Q}}$ to be in the intermediate model, we need to allow this so-called “mixing property”. And so, if we require it to hold for the conditions, we will need to require it to also hold for the automorphisms of $\dot {\mathbb {Q}}$ , etc., and so the idea itself is important. We can now proceed towards finding a combinatorial definition that will allow us to directly define the class of names, much like we did with ${\mathsf {HS}}$ in the case of a single symmetric extension.

Easily, every ${\mathscr {F}}$ -symmetric name is ${\mathscr {F}}$ -respected, and much like the definition of ${\mathscr {F}}$ -symmetric before it, this definition lends itself to a hereditary version. We will simply say “respected” when ${\mathscr {F}}$ is clear from context. If $\dot A$ carries an implicit structure (e.g., a forcing notion) then this structure is also required to be respected.

The idea is that being respected is “almost” being symmetric. We will soon weaken this property a bit further to accommodate the “mixing property” into the respected names.

Respect is the foremost necessary condition for developing a combinatorial characterisation of iterated symmetric extensions. Given a symmetric system $\langle {\mathbb {P}},{\mathscr {G}},{\mathscr {F}}\rangle $ and a ${\mathbb {P}}$ -name $\dot {\mathbb {Q}}$ , in order to define an automorphism of ${\mathbb {P}}\ast \dot {\mathbb {Q}}$ using some $\pi \in {\mathscr {G}}$ , the first thing we need to ensure is that $\pi $ respects $\dot {\mathbb {Q}}$ . Otherwise, $\langle p,\dot q\rangle \mapsto \langle \pi p,\pi \dot q\rangle $ is not an automorphism of ${\mathbb {P}}\ast \dot {\mathbb {Q}}$ .

Here we utilise the mixing property quite significantly, by defining $\dot \sigma \dot q$ to be the name guaranteed to be interpreted as the action of $\dot \sigma $ on the condition  $\dot q$ . If we were using Kunen iterations, we would have to define ${\textstyle \int _{\langle \pi ,\dot \sigma \rangle }}$ as a partial automorphism which is only defined when $p\mathrel {\Vdash }\dot \sigma \in {\mathscr {G}}$ . This is not a formal problem, but it makes the actual legwork a lot harder.

We will denote by ${\mathscr {G}}\ast \dot {\mathscr {H}}$ the group of all such automorphisms. In [Reference Karagila9] we refer to this group as the generic semidirect product. Turning our attention to the filters ${\mathscr {F}}$ and $\dot {\mathscr {K}}$ , we define a support to be $\langle \dot H_0,\dot H_1\rangle $ where both of them are ${\mathbb {P}}\ast \dot {\mathbb {Q}}$ -names such that $\mathrel {\Vdash }\dot H_0\in \check {\mathscr {F}}$ and $\dot H_1\in \dot {\mathscr {K}}$ . Note that we can always extend whatever condition to decide the actual group $\dot H_0$ , and likewise we can decide the actual ${\mathbb {P}}$ -name for $\dot H_1$ . However, using this approach allows us to take advantage of the mixing property.

We can now define the notion of respect relative to the supports. Namely, there is a pair $\langle \dot H_0,\dot H_1\rangle $ such that whenever $\langle p,\dot q\rangle \mathrel {\Vdash } \langle \check \pi ,\dot \sigma \rangle \in \langle \dot H_0,\dot H_1\rangle $ , which is to say $\langle p,\dot q\rangle \mathrel {\Vdash }\check \pi \in \dot H_0$ and $\dot \sigma \in \dot H_1$ , we have that $\langle p,\dot q\rangle \mathrel {\Vdash }{\textstyle \int _{\langle \pi ,\dot \sigma \rangle }}\dot A=\dot A$ . Whereas in Definition 2.4 we required that $\dot H_0$ would actually be a concrete group, here we allow a bit more leeway. If we consider this definition at a single-step forcing, we weakened the requirement that $H\in {\mathscr {F}}$ to requiring that $\dot H$ is a name for a group in ${\mathscr {F}}$ . If we mix names from ${\mathsf {HS}}_{\mathscr {F}}$ , then the result will be respected, since we can mix the groups, $\operatorname {\mathrm {sym}}(\dot x)$ , to obtain a name $\dot H$ which is forced to be in ${\mathscr {F}}$ .

A symmetric iteration (of length $\alpha$ ), or an iteration of symmetric extensions, is defined by specifying a sequence of names for symmetric systems, $\langle \dot {\mathbb {Q}}_\beta ,\dot {\mathscr {G}}_\beta , \dot {\mathscr {F}}_\beta \mid \beta <\alpha \rangle $ , and defining ${\mathbb {P}}_\alpha $ as the finite support iteration of the forcings $\dot {\mathbb {Q}}_\beta $ ; $\mathcal G_\alpha $ as the finite support generic semidirect product of the groups $\dot {\mathscr {G}}_\beta $ , made of sequences $\langle \dot \pi _\beta \mid \beta <\alpha \rangle $ such that and for all but finitely many $\beta $ , ; and $\mathcal F_\alpha $ as the collection of all supports, defined as follows.

We define supports as ${\mathbb {P}}_\alpha $ -names for sequences $\langle \dot H_\beta \mid \beta <\alpha \rangle $ such that $\mathrel {\Vdash }_\alpha \dot H_\beta \in \dot {\mathscr {F}}_\beta $ , and that $\mathrel {\Vdash }_\alpha \{\check \beta \mid \dot H_\beta \neq \dot {\mathscr {G}}_\beta \}^\bullet $ is finite. The last part is crucial, as it allows us the flexibility in “knowing something has a finite definition” while not committing to its specifics just yet.

Before we continue on, let us expand a bit on these definitions, with the caveat that the generality of the method is not needed for the Bristol model, as will be explained later, and so we encourage the interested reader to study the general framework for iterations as presented in [Reference Karagila9]. Given a sequence $\vec \pi $ in $\mathcal G_\alpha $ , the action on ${\mathbb {P}}_\alpha $ , denoted by ${\textstyle \int _{\vec \pi }}p$ , will be defined by recursion. This is easy if there is exactly one $\beta <\alpha $ for which . In that case ${\textstyle \int _{\vec \pi \restriction \beta }}=\operatorname {\mathrm {id}}$ and ${\textstyle \int _{\dot \pi _\beta }}$ is acting on ${\mathbb {Q}}_\beta $ as well as on the name of the quotient ${\mathbb {P}}_\alpha /{\mathbb {Q}}_{\beta +1}$ . And so,

$$\begin{align*}{\textstyle\int_{\vec\pi}}p=p\restriction\beta^\smallfrown\dot\pi_\beta(p(\beta))^\smallfrown\dot\pi_\beta(p\restriction(\beta,\alpha)).\end{align*}$$

Letting , then ${\textstyle \int _{\vec \pi }}$ is simply the composition ${\textstyle \int _{\dot \pi _{\beta _0}}}\dots {\textstyle \int _{\dot \pi _{\beta _n}}}$ .Footnote ¹¹ Looking at a support, $\vec H=\langle \dot H_\beta \mid \beta <\alpha \rangle $ , these do not quite define a subgroup of $\mathcal G_\alpha $ , however, for each $p\in {\mathbb {P}}_\alpha $ we can define a “local group” which is generated by $\{{\textstyle \int _{\vec \pi }}\mid \forall \beta <\alpha , p\mathrel {\Vdash }\dot \beta _\beta \in \dot H_\beta \}$ .Footnote ¹² This leads us to the definition of respect in this case. Namely, $\dot x$ is $\mathcal F$ -respected if there is a support $\vec H$ such that whenever $p\mathrel {\Vdash }\vec \pi \in \vec H$ , then $p\mathrel {\Vdash }{\textstyle \int _{\vec \pi }}\dot x=\dot x$ .

Let $\mathsf {IS}_\alpha $ be the class of ${\mathbb {P}}_\alpha $ -names which are hereditarily respected. This class is now the class of names which will be interpreted in the intermediate model of stage $\alpha $ . So to complete our definition of a symmetric iteration we need to require that $\langle \dot {\mathbb {Q}}_\alpha ,\dot {\mathscr {G}}_\alpha ,\dot {\mathscr {F}}_\alpha \rangle ^\bullet $ is in $\mathsf {IS}_\alpha $ , and indeed that it is respected by all automorphisms in $\mathcal G_\alpha $ , which guarantees that the action of $\mathcal G_\alpha $ extends to ${\mathbb {P}}_{\alpha +1}$ . We also have a forcing relation $\mathrel {\Vdash }^{\mathsf {IS}}_\alpha $ which is defined by relativising the names and quantifiers to $\mathsf {IS}_\alpha $ .

Using this definition we can show that when G is V-generic, then $\mathsf {IS}_\alpha ^G$ is a model of $\mathsf {ZF}$ , and if $\alpha =\beta +1$ , then this model is a symmetric extension of $\mathsf {IS}_\beta ^{G\restriction \beta }$ using the $\beta $ th symmetric system. On the other hand, if we want to continue a symmetric iteration, that is of course possible, but we need to make sure that the generic filter used is not only $\mathsf {IS}_\alpha ^G$ -generic, but rather $V[G]$ -generic, and we may need to shrink the groups $\dot {\mathscr {G}}_\beta $ in a few places to satisfy the requirement that each symmetric system is respected by the relevant $\mathcal G_\beta $ . This may present an issue if we want to continue our iteration by infinitely many steps, as we may need to shrink our groups infinitely many times, which might not be possible, as the filters of groups might not be sufficiently closed.

Here we arrive to our first obstacle when applying this definition to the Bristol model. We required the generic filter is V-generic for the entire iteration, whereas the Bristol model is defined inside a single Cohen real, so we need to find a way to modify the definition to allow for “pointwise genericity”, so that we can construct the Bristol model one step at a time, while finding “sufficiently generic” objects inside $L[c]$ . In the case of iterated forcing using pointwise genericity is not an issue for successor steps,Footnote ¹³ but here we run into the first problem we had defining the whole apparatus using mixing. The use of mixing allows us to use arbitrary antichains and predense sets to define the names in $\mathsf {IS}_\alpha $ , and so we end up with names in $\mathsf {IS}_\alpha $ which encode some generic information in them. This means that pointwise generic filters might not be able to correctly interpret such names, since they might not actually meet the relevant antichains.

The “easy” way to leave this mess is to require that we relativise the definition pointwise. That is, at each step we only take names which are in the intermediate model. But this adds a layer of complexity when defining the actual forcing ${\mathbb {P}}_\alpha $ , or the automorphisms in $\mathcal G_\alpha $ , or using these in the same manner that we are used to when working with symmetric extensions. Even worse, while for finite support iterations all of these different constructions are equivalent, this is not the case if we want to extend our definition to other types of iterations.

We take a different route instead. Looking at products as a type of degenerate iterations, we may want to mimic this definition here. But a copy–paste approach is bound to result in just a product of symmetric extensions. While this is fine, it is not what we are looking for. We want to force over the symmetric extensions, but with a “very canonically defined forcing”. The idea is that we want to iterate ${\mathbb {P}}\ast \dot {\mathbb {Q}}$ , and in $V[G]$ , where $G\subseteq {\mathbb {P}}$ , the forcing $\dot {\mathbb {Q}}^G$ is isomorphic to a forcing in V, and to some extent, this isomorphism does not even depend on G. This means that we are really taking a product. But in the symmetric extension given by ${\mathbb {P}}$ , the forcing $\dot {\mathbb {Q}}^G$ is not isomorphic to any partial order in V, maybe because it cannot be well-ordered, or maybe due to a similar consideration. Nevertheless, it is distinct from forcings in V as far as ${\mathsf {HS}}^G$ is concerned.

A symmetric iteration is called productive if it behaves, essentially as a product. Namely, each $\dot {\mathbb {Q}}_\alpha $ , $\dot {\mathscr {G}}_\alpha $ , and $\dot {\mathscr {F}}_\alpha $ are $\bullet $ -names, with deciding all the relevant formulas. Namely, decides the truth value of when two conditions, automorphisms or groups, are equal; it decides when $\dot \pi \dot q=\dot q'$ , etc. In this case, the symmetric system is “essentially a copy of a ground model system”,Footnote ¹⁴ and the iteration can be presented as a bona fide product of these names. For this concept to be complete we also need to remove the flexibility in the definition of supports, which is needed for the iterative definition of $\mathsf {IS}_\alpha $ . We require, in the productive case, that $\langle \dot H_\beta \mid \beta <\alpha \rangle $ is an excellent support, meaning that $\dot H_\beta $ is a name appearing in $\dot {\mathscr {F}}_\beta $ , and in particular a ${\mathbb {P}}_\beta $ -name, and the finite set of non-trivial coordinates is decided in advance. This is in line with the previous demands: everything is decided in advance, this is “almost a product”.

Now that we have restricted the iteration, we can extend the concept of “generic”.

It turns out that symmetrically generic filters are exactly the filters needed to interpret symmetric names, and that the following theorem holds.

It is not hard to check now that at least for the successor steps, the iteration of “pointwise” symmetrically generic filters is indeed a symmetrically generic filter.

We finish this overview with a preservation theorem.

Theorem 2.8. Suppose that $\langle \dot {\mathbb {Q}}_\alpha ,\dot {\mathscr {G}}_\alpha ,\dot {\mathscr {F}}_\alpha \mid \alpha <\delta \rangle $ defines a symmetric iteration, and G is V-generic for the iteration. Moreover, assume that for every $\alpha <\delta $ , is weakly homogeneous and $\dot {\mathscr {G}}_\alpha $ is rich enough to witness this.Footnote ¹⁷ Let $\eta $ be an ordinal such that there exists $\alpha _0<\delta $ that for any $\alpha \in (\alpha _0,\delta )$ the following equality holds:

$$\begin{align*}V_\eta^{\mathsf{IS}_\alpha^{G\restriction\alpha}}=V_\eta^{\mathsf{IS}_{\alpha+1}^{G\restriction\alpha+1}}.\end{align*}$$

Then $V_\eta ^{\mathsf {IS}_\delta ^G}=V_\eta ^{\mathsf {IS}_{\alpha _0+1}^{G\restriction \alpha _0+1}}$ . In other words, if no sets of rank ${<}\eta $ were added at successor steps, none were added at limit steps either.

This theorem is in stark contrast to the familiar case in the usual context of iterated forcing: iterating, with finite support, forcings which are not c.c.c. will collapse cardinals; and iterating non-trivial forcings, even if they are c.c.c., will add Cohen reals at limit steps. But in the case of symmetric iterations, even if the forcings are non-trivial, as long as they are homogeneous and do not add reals, the limit steps will not add reals either.

As a consequence, we can extend our apparatus now to an $\mathrm {Ord}$ -length iteration while preserving $\mathsf {ZF}$ in the resulting model. Moreover, the result holds for productive iterations with symmetrically generic filters, as one can state it in the language of forcing, rather than talking about $V_\eta $ of various models. See also Section 9.2 of [Reference Karagila9].

2.4 Permutable families and scales

The key mechanism in the construction of the Bristol model is “decoding a long sequence from a short sequence”. This can mean a sequence of length $\omega _1$ from a Cohen real, or a sequence of length $\omega _{43}$ from one of length $\omega _{42}$ . We use almost disjoint families in successor steps to repeatedly decode these sequences, and we use a particular type of a PCF-scale to succeed at this task when we are at limit steps. This will be as good a place as any to remind the reader that we are working in $\mathsf {ZFC}$ , especially when thinking about these combinatorial objects that are used here.

Here we use $=^*$ to mean equality up to a bounded subset of $\kappa $ . Which $\kappa $ , of course, will be clear from the context, so we will spare the notation $=^*_\kappa $ , or worse, from the reader.

We are looking for an abstract property of an almost disjoint family which will ensure that it implements any bounded permutation of $\kappa ^+$ , that is, any permutation of $\kappa ^+$ which is the identity on a tail can be implemented.

Definition 2.10. Let $\kappa $ be a regular cardinal, $\{A_\alpha \mid \alpha <\kappa ^+\}\subseteq [\kappa ]^\kappa $ is called a permutable family if for $\alpha \neq \beta $ , $\sup (A_\alpha \cap A_\beta )<\kappa $ , and for every $I\in [\kappa ^+]^{<\kappa ^+}$ there is a pairwise disjoint family $\{B_\xi \mid \xi \in I\}$ such that $B_\xi =^*A_\xi $ and

$$\begin{align*}\alpha\in I\iff A_\alpha\cap\bigcup_{\xi\in I} B_\xi\text{ is unbounded in }\kappa.\end{align*}$$

We call $\{B_\xi \mid \xi \in I\}$ as in this definition a disjoint approximation, and in the case where $B_\xi \subseteq A_\xi $ , we say it is a disjoint refinement. Note that both of these always exist, as long as $|I|\leq \kappa $ , by a standard transfinite recursion argument using the regularity of $\kappa $ .

Having fixed a permutable family, if $\pi \colon \kappa \to \kappa $ implements $\Pi $ , we will denote this by $\iota (\pi )=\Pi $ .

While these definitions are given for regular cardinals, we will only use them in the basis case and successor case. For the limit case, where $\kappa $ is a limit cardinal, we need to use a slightly different machinery, as the goal is to coalesce the information from previous steps and use it as a kind of “short sequence”. In some way, inaccessible cardinals are the “simpler case” compared to singular cardinals. Nevertheless, there is no need of separating the two.

Definition 2.15. Let $\lambda $ be a limit cardinal, and let $\operatorname {\mathrm {SC}}(\lambda ) = \{\mu ^+\mid \mu <\lambda \}$ . Let $\{f_\alpha \mid \alpha <\lambda ^+\}$ be a scale in $\prod \operatorname {\mathrm {SC}}(\lambda )$ .Footnote ¹⁸ Given a sequence of permutations $\vec \pi =\langle \pi _\theta \mid \theta \in \operatorname {\mathrm {SC}}(\lambda )\rangle $ such that $\pi _\theta $ is a permutation of $\theta $ , we say that $\vec \pi $ implements a function $\Pi \colon \lambda ^+\to \lambda ^+$ if for every $\alpha <\lambda ^+$ and for every large enough $\theta $ ,

$$\begin{align*}f_{\Pi(\alpha)}(\theta) = \pi_\theta f_\alpha(\theta).\end{align*}$$

We call a scale permutable if it implements every bounded permutation of  $\lambda ^+$ .

As with the case of permutable families, we denote by $\iota (\vec \pi )$ the permutation $\Pi $ that is implemented by $\vec \pi $ .

The fact that inaccessible cardinals are the “simpler case” can be trivially seen as a consequence of the following proposition, while remembering that working in $V=L$ we always have the wanted cardinal arithmetic.

This can be shown by a simple transfinite recursion, in a very similar fashion to the permutable family case.

The proof of this proposition can be found as Theorem 3.27 in [Reference Karagila8]. The idea of the proof goes back to the Bristol group, and utilises the work of Cummings, Foreman, and Magidor in [Reference Cummings, Foreman and Magidor2] where it is shown that $\square ^*_\lambda $ implies the existence of “better scales”. The aforementioned Theorem 3.27 shows that a better scale is in fact permutable.

The key point in proving a better scale is a permutable scale is that given any $I\in [\lambda ^+]^{<\lambda ^+}$ , we can find a function $d\colon I\to \operatorname {\mathrm {SC}}(\lambda )$ such that $\{f_\alpha "[d(\alpha ),\lambda )\mid \alpha \in I\}$ is a family of pairwise disjoint sets. The proof of Theorem 3.27 also shows that even if we are only allowing $\pi _\theta $ to be a bounded permutation of $\theta $ , this is still enough to ensure that we can implement every bounded permutation of $\lambda ^+$ using a permutable scale. We say that a sequence of permutation groups of each $\theta \in \operatorname {\mathrm {SC}}(\lambda )$ is rich enough if we can require $\pi _\theta $ to be in the relevant group when we find an implementing sequence.

Definition 2.18. Let $\lambda $ be a limit cardinal, and for every $\theta \in \operatorname {\mathrm {SC}}(\lambda )$ , let ${\mathscr {G}}_\theta $ be a rich enough group of permutations of $\theta $ . The derived group is the subgroup ${\mathscr {G}}$ of the full support product $\prod _{\theta \in \operatorname {\mathrm {SC}}(\lambda )}{\mathscr {G}}_\theta $ consisting of all sequences $\vec \pi =\langle \pi _\theta \mid \theta \in \operatorname {\mathrm {SC}}(\lambda )\rangle $ which implement a bounded permutation of $\lambda ^+$ .

For $\eta <\lambda ^+$ and $f\in \prod \operatorname {\mathrm {SC}}(\lambda )$ we let $K_{\eta ,f}$ be the group

$$\begin{align*}\{\vec\pi\in{\mathscr{G}}\mid\iota(\vec\pi)\restriction\eta=\operatorname{\mathrm{id}}\text{ and for all }\theta\in\operatorname{\mathrm{SC}}(\lambda), \pi_\theta\restriction f(\theta)=\operatorname{\mathrm{id}}\}.\end{align*}$$

The derived filter is the filter generated by $\{K_{\eta ,f}\mid \eta <\lambda ^+,f\in \prod \operatorname {\mathrm {SC}}(\lambda )\}$ .

We can weaken the definition of $K_{\eta ,f}$ and replace $\eta $ by a bounded subset of $\lambda ^+$ , i.e., $I\in [\lambda ^+]^{<\lambda ^+}$ , and replace f by a sequence of bounded sets of each $\theta $ . But as the definition is complicated enough as it is, it is easier to just use $\eta $ and f as upper bounds.

3.1 Example: First steps

Let ${\mathbb {P}}$ be the Cohen forcing, and in this case we mean $p\in {\mathbb {P}}$ is a function from a finite subset of $\omega $ into $2$ . Let us omit the index from $\mathcal A_0$ , as we are only concerned with the first step at the moment, so $A_\alpha $ denotes the $\alpha $ th set in the first permutable family. We let ${\mathscr {G}}$ and ${\mathscr {F}}$ denote the derived group and filter from $\mathcal A$ . The action of ${\mathscr {G}}$ on ${\mathbb {P}}$ is the natural one:

$$\begin{align*}\pi p(\pi n)=p(n).\end{align*}$$

We denote by $\dot c$ the canonical name for the Cohen real. For $A\subseteq \omega $ let ${\mathbb {P}}\restriction A$ denote the subforcing $\{p\in {\mathbb {P}}\mid \operatorname {\mathrm {dom}} p\subseteq A\}$ , and let $\dot c_A$ denote the name

$$\begin{align*}\{\langle p,\check n\rangle\mid p(n)=1\land\operatorname{\mathrm{dom}} p\subseteq A\}.\end{align*}$$

Of course, $\dot c_A$ is the canonical name of the generic real added by ${\mathbb {P}}\restriction A$ . We have now that $\pi \dot c_A=\dot c_{\pi "A}$ . In general we say that a name $\dot x$ for a set of ordinals is decent if every name appearing in it is of the form $\check \xi $ for some $\xi \in \mathrm {Ord}$ . We say that a name for a set of ordinals is an A-name if it is a ${\mathbb {P}}\restriction A$ -name.

Proof Let $\mathcal B$ be a disjoint approximation such that $\operatorname {\mathrm {fix}}(\mathcal B)\subseteq \operatorname {\mathrm {sym}}(\dot x)$ and define $\dot x_*$ as

$$\begin{align*}\dot x_* = \left\{\langle p,\check n\rangle\mathrel{}\middle|\mathrel{} p\mathrel{\Vdash}\check n\in\dot x\land\operatorname{\mathrm{dom}} p\subseteq\bigcup\mathcal B\right\}.\end{align*}$$

It is clear that , to show equality it is enough to prove that if $p\mathrel {\Vdash }\check n\in \dot x$ , then $p\restriction \bigcup \mathcal B\mathrel {\Vdash }\check n\in \dot x$ . Let $q\leq p\restriction \bigcup \mathcal B$ .

By the very definition of a disjoint approximation $\bigcup \mathcal B$ is co-infinite, so we may find a finite set E such that

$$\begin{align*}E\cap (\bigcup \mathcal B\cup \operatorname {\mathrm {dom}} p)=\varnothing \text{ and } |E|=|\operatorname {\mathrm {dom}} q\setminus \bigcup \mathcal B|.\end{align*}$$

Then let $\pi $ be a permutation which extends a bijection between the two sets $\operatorname {\mathrm {dom}} q\setminus \bigcup \mathcal B$ and E, and is the identity elsewhere. Being a finitary permutation it implements the identity function, so indeed $\pi \in {\mathscr {G}}$ , and by its very definition $\pi \in \operatorname {\mathrm {fix}}(\mathcal B)$ , so $\pi \dot x=\dot x$ . So if q had forced $\check n\notin \dot x$ , we would have $\pi q\mathrel {\Vdash }\pi \check n\notin \pi \dot x$ , which is the same as $\pi q\mathrel {\Vdash }\check n\notin \dot x$ . Alas, $\pi q$ and p are clearly compatible, and so this is impossible.

We let G be a V-generic filterFootnote ²⁰ and let M denote the symmetric extension ${\mathsf {HS}}^G$ , as is standard, we will “omit the dot” to indicate the interpretation of a name, so c is going to be $\dot c^G$ , etc. The following is a very easy corollary from the above proposition.

This is where we start seeing the importance of the permutable family, as opposed to any almost disjoint family. If $\{X_\alpha \mid \alpha <\omega _1\}$ is an almost disjoint family, then $\{c\cap X_\alpha \mid \alpha <\omega _1\}$ is a family of mutually generic Cohen reals, any finitely many of them are mutually generic over any other finite subfamily. But in our case, where the almost disjoint family is in fact a permutable one we get countable mutual genericity. Any countable subset of $\{c\cap A_\alpha \mid \alpha <\omega _1\}$ is simultaneously mutually generic over any other countable subset (provided they are pairwise disjoint).

We can actually prove more. Nothing in the proof of Proposition 3.2 will change if we assume that $\dot x$ is a name of an arbitrary set of ordinals. This shows that every set of ordinals lies in an intermediate model given by $c\cap \bigcup \mathcal B$ for some disjoint approximation. Another way to see this fact is to note that $L[c]$ is, after all, only a Cohen extension by a single real. So every set of ordinals is constructible from a single real.

Of course, we can prove directly that $\mathbb R^M$ cannot be well-ordered in M, and this argument can be found in the proof of Theorem 2.7 in [Reference Karagila8].

We can see Proposition 3.2 as somehow indicating not only that the reals of M are generated by countable parts of $\mathcal A$ , but in fact if $\langle T_\alpha \mid \alpha <\omega _1\rangle $ is a tower generated by $\mathcal A$ , then we actually have that $\mathbb R^M$ is the increasing union of $\mathbb R^{L[c\cap T_\alpha ]}$ . This was the original approach of the Bristol group.

At this point, one might expect that the decoded sequence would be $\langle c\cap A_\alpha \mid \alpha <\omega _1\rangle $ or somehow $\langle c\cap T_\alpha \mid \alpha <\omega _1\rangle $ . But of course, this is not the case. For starters, we want to somehow “fuzzy out” some of the information as to guarantee that c is not constructible from the sequence. So instead of $c\cap A_\alpha $ , we will look at $\mathbb R^{L[c\cap A_\alpha ]}$ . But more importantly, the decoded sequence is not even in M. Indeed, if we want this sequence to play the role of the Cohen real in the next step, that means that it needs to be forced into M instead.

Let us begin by understanding $\mathbb R^{L[c\cap A_\alpha ]}$ . In what way does this set “fuzzy out” some information? Well, for one, $c\cap A_\alpha $ is not the obvious real from which we construct this model. Indeed, any finite modification would work, and much more. In fact, any $\pi \in {\mathscr {G}}$ for which $\iota (\pi )(\alpha )=\alpha $ will satisfy that $\pi \dot c_{A_\alpha }$ is a name generating the same set of reals, as it is $c\cap A_\alpha $ up to a permutation of $A_\alpha $ and a finite set.

We say that a name $\dot x$ is an almost A-name if there exists B such that $A=^*B$ and $\dot x$ is a B-name. We now define

$$\begin{align*}\dot R_\alpha=\{\dot x\mid\dot x\text{ is a decent almost }A_\alpha\text{-name}\}^\bullet.\end{align*}$$

Similar arguments as we have seen so far also prove the following statement.

And here we arrive to the key point. Let $\varrho $ denote $\langle R_\alpha \mid \alpha <\omega _1\rangle $ and let $\dot \varrho $ denote its name. We will also write $\varrho _I$ and $\dot \varrho _I$ for the restriction of the sequence to I, similar to $\dot c_A$ .Footnote ²¹ Indeed, $\varrho $ is going to be the decoded sequence, and it is of course going to be M-generic, but we need to find a suitable partial order. Proposition 3.6 provides us with a good clue: every initial segment of this sequence is in fact in M, so we can “safely” approximate this sequence.

It will be somewhat more convenient to use subsets of $\omega _1$ , as we did with the Cohen forcing, rather than proper initial segments. This makes it easier to talk about A-names and almost A-names. And again for convenience (and so we can claim productivity, of course), we are also going to limit ourselves to subsets of $\omega _1$ which are already in L.

In other words, $\varrho $ is M-generic for ${\mathbb {Q}}$ . The idea behind ${\mathbb {Q}}$ is that we want the smallest “reasonable” set which contains our generic filter (i.e., partial approximations of $\varrho $ ), and the easiest way to do that is to simply apply all permutations and obtain a set. But the true intuition behind ${\mathbb {Q}}$ , and really behind the whole decoding apparatus, comes from understanding $\pi \varrho _I$ .

The model M knows of the set $R=\{R_\alpha \mid \alpha <\omega _1\}$ , it just does not know a well-ordering of this set. And we are trying to remedy that. As we know already every $\pi \in {\mathscr {G}}$ implements a permutation of $\omega _1$ , which induces a permutation of R moving countably many points. So $\pi $ shuffles R and thus modifies the range of $\varrho _I$ . But R is an extremely impoverished set as far as M is concerned. This is not particularly important for the construction, and can be skipped entirely, but it is an interesting fact.

Proof Suppose that $\dot X,\dot Y\in {\mathsf {HS}}$ and $p\mathrel {\Vdash }^{{\mathsf {HS}}}"\dot X,\dot Y\subseteq \dot R$ and are uncountable”. Let $\mathcal B$ be a disjoint approximation such that $\operatorname {\mathrm {dom}} p\subseteq \bigcup \mathcal B$ and such that $\operatorname {\mathrm {fix}}(\mathcal B)\subseteq \operatorname {\mathrm {sym}}(\dot X)\cap \operatorname {\mathrm {sym}}(\dot Y)$ . By uncountability, we can extend p to some q for which there are $\alpha ,\beta $ such that:

1. (1) $q\mathrel {\Vdash }^{{\mathsf {HS}}} \dot R_\alpha \in \dot X$ and $\dot R_\beta \in \dot Y$ .

2. (2) $A_\alpha \cap \bigcup \mathcal B$ and $A_\beta \cap \bigcup \mathcal B$ are finite.

By enlarging one of the sets in $\mathcal B$ , if necessary, we can also assume that $\operatorname {\mathrm {dom}} q\subseteq \bigcup \mathcal B$ as well. We can now find a permutation $\pi \in \operatorname {\mathrm {fix}}(\mathcal B)$ such that $\iota (\pi )$ is the $2$ -cycle switching $\alpha $ and $\beta $ .

Applying $\pi $ to the first property of q we have that

$$\begin{align*}\pi q\mathrel {\Vdash }^{{\mathsf {HS}}}\pi \dot R_\alpha \in \pi \dot X,\pi \dot R_\beta \in \pi \dot Y.\end{align*}$$

But since $\pi \in \operatorname {\mathrm {fix}}(\mathcal B)$ we have that $\pi q=q$ and $\pi \dot X=\dot X$ and $\pi \dot Y=\dot Y$ . This means that $q\mathrel {\Vdash }\dot R_\beta \in \dot X$ and $\dot R_\alpha \in \dot Y$ . In particular q forces that $\dot X$ and $\dot Y$ are not disjoint.

Next we want to prove that every partition of R is almost entirely singletons. We will only sketch the idea behind the argument. If $\dot S\in {\mathsf {HS}}$ and $p\mathrel {\Vdash }^{{\mathsf {HS}}}"\dot S$ is a partition of $\dot R$ into uncountably many cells”, let $\mathcal B$ be an approximation such that $\operatorname {\mathrm {fix}}(\mathcal B)\subseteq \operatorname {\mathrm {sym}}(\dot S)$ . Pick $\alpha ,\beta $ as above, so that we may switch between them without interfering with $\mathcal B$ , and we can implement the $2$ -cycle $(\alpha \ \beta )$ without changing any given condition. This means that any point that was in the same cell as $\alpha $ must have moved to the cell containing  $\beta $ . But we only moved two points, so $\alpha $ and $\beta $ must have been isolated as singletons. And therefore the only non-singleton cells come from a partition of $\mathcal B$ itself.

Getting back to the matter at hand, we want to prove that $\varrho $ is M-generic. Namely, if $D\subseteq {\mathbb {Q}}$ is a dense subset and $D\in M$ , we want to prove that there is some $\alpha <\omega _1$ such that $\varrho _\alpha \in D$ . In [Reference Karagila8] we prove this by proving a technical lemma about names of dense open sets, Lemma 2.12.Footnote ²² In the article we use this lemma also as a means for proving that ${\mathbb {Q}}$ is $\sigma $ -distributive, and therefore does not add any new reals to the model (and as a corollary, it does not force $\mathsf {AC}$ back into the universe somehow), which also finishes the proof that $\varrho $ is indeed the sequence we are looking for.

We will prove the genericity of $\varrho $ using a simplified version of Lemma 2.12, and provide a separate argument for the distributivity.

Proof Let $\dot D$ be a name as above, and let $\mathcal B$ be a disjoint approximation such that $\operatorname {\mathrm {fix}}(\mathcal B)\subseteq \operatorname {\mathrm {sym}}(\dot D)$ . Let $p\mathrel {\Vdash }"\dot D$ is a dense open subset of $\dot {\mathbb {Q}}$ ”. Let $\alpha $ be large enough such that if $A_\xi \cap \bigcup \mathcal B$ is infinite, then $\xi <\alpha $ . Our strategy is to find “enough” extensions of $\dot \varrho _\alpha $ so that one of them will be both $\dot \varrho _\eta $ , and in $\dot D$ .

First we prove the following claim: suppose that $q\leq p$ and $q\mathrel {\Vdash }\pi \dot \varrho _A\in \dot D$ , where $\alpha \subseteq A$ and $\iota (\pi )\restriction \alpha =\operatorname {\mathrm {id}}$ , then $q\mathrel {\Vdash }\dot \varrho _A\in \dot D$ . In other words, if $\pi \dot \varrho _A$ is an extension of $\dot \varrho _\alpha $ which lies in $\dot D$ , then $\dot \varrho _A$ is in $\dot D$ as well. Of course, this is true because there is some $\tau \in \operatorname {\mathrm {fix}}(\mathcal B)$ such that $\iota (\tau )=\iota (\pi )^{-1}$ and $\tau q=q$ .

As $\pi $ implemented the identity up to $\alpha $ , $\iota (\pi )^{-1}$ will also be the identity up to $\alpha $ , and thus we can implement it using an automorphism in $\operatorname {\mathrm {fix}}(\mathcal B)$ which fixes q, since $\operatorname {\mathrm {dom}} q$ is a finite set.

This completes the proof of the claim since

$$\begin{align*}\tau q=q\mathrel{\Vdash}\tau\pi\dot\varrho_A=\dot\varrho_{\iota(\tau)\circ\iota(\pi)"A}=\dot\varrho_A\in\tau\dot D=\dot D.\end{align*}$$

In turn, this is enough to prove the genericity of $\varrho $ : find a maximal antichain below p of conditions which decide some $\pi \dot \varrho _A$ as above, and by openness we can assume that each such A is in fact an ordinal. Let $\eta $ be the supremum of this countable set of ordinals, and we have that $p\mathrel {\Vdash }\dot \varrho _\eta \in \dot D$ .

The final claim is that ${\mathbb {Q}}$ does not add new reals to M. This, as we remarked, ensures that $M[\varrho ]\neq L[c]$ . It has an added effect that ${\mathbb {Q}}$ is not adding any new sets of ordinals, or any countable sequences of ground model objects. In [Reference Karagila8] the proof utilised a stronger version of the above proof which lets us intersect a countable sequence of dense open sets. Here we take a slightly different approach.Footnote ²³

3.2 Outline of successor steps

The first two steps are fairly indicative of the standard successor step. The main change, of course, is that we need to understand what replaces $\mathbb R$ , L, and c. We have a hint as to what replaces c, namely, the sequence $\varrho $ that we ended up with after forcing with ${\mathbb {Q}}$ . We also have an idea on what to replace $\mathbb R$ with, that would be $\mathcal {P}(\mathbb R)$ , and L is to be replaced by M itself.

We can make this much clearer if we recast the example above by replacing $\mathbb R$ with $V_{\omega +1}$ . We can also replace c with a sequence of elements of $V_\omega $ , of course, but this seems to needlessly complicate things. After all, the case of $\omega $ is separate anyway.

For a more uniform approach, we denote by $M_\alpha $ the $\alpha $ th step in the construction, which is a model of $\mathsf {ZF}$ intermediate between L and $L[c]$ . We will also write $\varrho _\alpha $ to denote the generic sequence for ${\mathbb {Q}}_\alpha $ , the $\alpha $ th forcing. So ${\mathbb {Q}}_0$ is Cohen forcing and $\varrho _0$ is c itself.

At each successor step we have $M_{\alpha +1}$ defined from $\varrho _\alpha $ , which was the $M_\alpha $ -generic sequence for ${\mathbb {Q}}_\alpha $ . We will assume that while $\varrho _\alpha \notin M_{\alpha +1}$ , for every $\xi <\omega _{\alpha +1}$ , $\varrho _\alpha \restriction A_\xi ^\alpha \in M_{\alpha +1}$ , where $A^\alpha _\xi $ is the $\xi $ th member of the permutable family we fixed in advance. Moreover, for every $I\in [\omega _{\alpha +1}]^{<\omega _{\alpha +1}}\cap L$ , the sequence $\langle \varrho _\alpha \restriction A^\alpha _\xi \mid \xi \in I\rangle $ is in $M_{\alpha +1}$ .

We now want to define ${\mathbb {Q}}_{\alpha +1}$ and $\varrho _{\alpha +1}$ . For this we replace $\mathbb R^{L[c\cap A_\xi ]}$ that we had in the first step with $V_{\omega +\alpha +2}^{M_{\alpha +1}[\varrho _\alpha \restriction A_\xi ^\alpha ]}$ , let us denote this as $R_\xi $ for now. The rest is more or less the same as above, relying, of course, on the recursive fact that any previous $M_\beta $ were defined much in the same way as we are defining ${\mathbb {Q}}_{\alpha +1}$ , $\varrho _{\alpha +1}$ , and $M_{\alpha +2}$ .

Let ${\mathbb {Q}}_{\alpha +1}$ denote the set of approximations of $\varrho _{\alpha +1}=\langle R_\xi \mid \xi <\omega _{\alpha +1}\rangle $ whose domains are in L. We are being vague, of course, as to what counts as “approximation” in this context. The idea is that we may permute the different $R_\xi $ amongst themselves using a permutation of $\omega _{\alpha +1}$ which is coming from L.

The lemmas in the general case are exactly the same as we had before. The genericity of $\varrho _{\alpha +1}$ is proved by the same argument as Proposition 3.10, and while the distributivity argument is also similar, it is worth writing down.

Finally, we define $M_{\alpha +2}$ as the symmetric extension obtained by applying the derived group and filter using the permutable family $\mathcal A_{\alpha +1}$ . If we now consider $V_{\omega +\alpha +2}^{M_{\alpha +1}[\varrho _{\alpha +1}\restriction A^{\alpha +1}_\xi ]}$ , this set is in $M_{\alpha +2}$ . Indeed, every proper initial segment of $\varrho _{\alpha +2}$ is in $M_{\alpha +2}$ , where $\varrho _{\alpha +2}$ is the sequence of these $V_{\omega +\alpha +2}$ , and it has length $\omega _{\alpha +2}$ . We can now show that it is $M_{\alpha +2}$ -generic, etc., and thus all shall prosper.

The successor case can be found in [Reference Karagila8] as Section 4.4, as well as Sections 4.7 and 4.10 for the successor of limit iterands. We are not separating the successor of limit steps in this text as the idea is the same with very minor variations, so as far as outlines go, there is essentially no difference.

3.3 Outline of limit steps

The limit step is divided into two parts. We have to contend with the iteration at a limit step, i.e., the finite support limit of the previous steps, and we have to deal with the limit step itself. An observant reader will notice that we did not utilise the full power of the framework of iterating symmetric extensions until now. Indeed, as far as successor steps are concerned any automorphism coming from coordinates before $\alpha $ itself will implement the identity function on the $(\alpha +1)$ th iterand, rendering it moot.

It is here, at the limit, where we need to utilise the machinery as a whole. In fact, this machinery will do most of the heavy lifting at this stage. By Proposition 3.12 we have that the rank initial segments of the universe are stabilising, indeed for any $\beta>\alpha $ we have $V_{\omega +\alpha +1}^{M_{\alpha +1}}=V_{\omega +\alpha +1}^{M_\beta }$ . The work left at this stage is making sure that the generic sequences we collected thus far are symmetrically generic, and setting up the stage for the limit step iterand. So it is a good idea to understand the limit step as a whole before proceeding to the details.

The main idea is that limit steps coalesce the information we have up to that point. Arriving to the limit is easy, as we said, the machinery of productive iterations is working for us there. But how do we proceed now? We are limited by two factors that we need to ensure continue to hold when we deal with the $\alpha $ th step:

1. (1) $V_{\omega +\alpha }$ is stable. That is, no new sets of low rank are added, and

2. (2) whatever we do is coherent with the other limit steps.

One simple way of ensuring this is by taking the products of previous successor steps. This way, if $\alpha <\beta $ are two limit ordinals, then ${\mathbb {Q}}_\alpha $ is going to be, in some sense, a rank initial segment of ${\mathbb {Q}}_\beta $ . But we can think about this from a different angle.

At each step, we gathered $V_{\omega +\xi }$ of various intermediate models, for $\xi <\alpha $ , our limit ordinal. But these are smoothed out, in a sense, as we progress up the hierarchy, as each $V_{\omega +\xi +1}^{M_{\xi +2}}$ contains each $V_{\omega +\xi }^{M_{\xi +1}}$ . But what if we could pick just one sequence, and remember it? In that case we are not going to add bounded sets to $V_{\omega +\alpha }$ , at least not if we are being careful, and instead we only add this sequence. This idea should seem somewhat familiar to readers of all walks of set theory. After all, if we want to add a new subset to $\aleph _\omega $ , it is easy to add Cohen subsets to each $\aleph _n$ first, and then choose a point from each one, creating a new cofinal sequence.Footnote ²⁴ Similar ideas, in one way or another, show up through Prikry-style forcings as well.

The coherence of limit steps has another very important use for the limit iterand. The following definition is very important as well.

So we want for the limit step that the iteration ${\mathbb {P}}_\alpha $ can move about conditions in ${\mathbb {Q}}_\alpha $ . If ${\mathbb {P}}_\alpha $ is a finite support iteration, then either ${\mathbb {Q}}_\alpha $ needs to have some finitary flavour to its conditions, or somewhere along the iteration we had to condense the conditions from finitary to infinitary. Indeed, this is the very meaning of “coalescing” the successor steps.

To sum up, at the limit step of the iteration we use the properties of the iteration so far to ensure that the rank initial segments of the models stabilise, and indeed that the sequence of generic sequences is symmetrically generic for the iteration. We then want to have a forcing such that the sequences which lie in the product of the successor-step generics combine to form a generic for it, here the permutable scales will come in naturally, as we are concerned with products of increasingly longer sequences modulo the bounded ideal.

We return to the context of the Bristol model’s construction. We denote by ${\mathbb {P}}_\alpha $ the iteration up to $\alpha $ and by ${\mathbb {Q}}_\alpha $ the $\alpha $ th iterand, as we did before, and for now we will assume that those iterands were defined also for limit steps. If this proves to be somewhat confusing, the section can be read twice, first assuming $\alpha =\omega $ .

We have seen this for the case of $\alpha $ being $0$ or a successor.Footnote ²⁶ We will see the rest of the cases in this section as we progress through it. But for now it is easier to take this as a working assumption.

This is essentially Proposition 4.4 ( $\alpha =\omega $ ) and Proposition 4.15 ( $\alpha $ an arbitrary limit ordinal) in [Reference Karagila8]. We will prove this statement in Section 5.2, as the original proof had a minor gap that needs to be corrected anyway.

But this means that we can understand the limit iterand fairly well now. We know that for $\beta <\alpha $ , $V_{\omega +\beta +1}^{M_\beta }=V_{\omega +\beta +1}^{M_\alpha }$ , and that not only we have a model of $\mathsf {ZF}$ which lies within $L[c]$ , but that it is in fact an iteration of symmetric extensions, which means that we understand exactly the objects which lie within it and the truth value of statements about these objects from a forcing-theoretic point of view. We are now free to examine the iterand  ${\mathbb {Q}}_\alpha $ .

As we reiterate time and time again, we want to ensure that no sets of rank $\omega +\alpha $ are added. In the successor steps we did that by making sure that ${\mathbb {Q}}_\alpha $ is sufficiently distributive. For the limit step this will pose a problem. If we are to continue with our successor steps, then the $(\alpha +1)$ th step needs to have a sequence of length $\omega _{\alpha +1}$ . But without adding any sequences of length $\omega _\alpha $ , this would mean that the sequence must have “mostly existed” already. So the forcing cannot be ${<}\omega _\alpha $ -distributive, let alone ${\leq }|V_{\omega +\alpha }|$ -distributive. In fact, if our plan is to add sequences of length $\alpha $ of sets of the form $V_{\omega +\beta }$ of some inner model, then at stages where $\operatorname {\mathrm {cf}}(\alpha )=\omega $ we must have added an $\omega $ -sequence.

The solution, as it turns out, is to not be distributive at all, but ensure that after applying the symmetric part of the step (rather than just the generic extension) we managed to remove any new set in $V_{\omega +\alpha }$ . So even if we do not have a distributive forcing, we at least preserve the rank initial segments of the universe.

We define $\varrho _\alpha $ , where $\alpha $ is a limit, as a “copy of $\prod \operatorname {\mathrm {SC}}(\omega _\alpha )^L$ ”. This means that for every $f\in \prod \operatorname {\mathrm {SC}}(\omega _\alpha )^L$ we define $\varrho _{\alpha ,f}$ to be $\langle \varrho _{\beta +1}(f(\beta +1))\mid \beta <\alpha \rangle $ , and $\varrho _\alpha $ is defined as $\langle \varrho _{\alpha ,f}\mid f\in \prod \operatorname {\mathrm {SC}}(\omega _\alpha )^L\rangle $ .

How should we define ${\mathbb {Q}}_\alpha $ , then? The idea is always “bounded approximations”, but having the virtue of being a “two-dimensional object” this means that boundedness has two sides to it. Let us first deal with the one-dimensional counterparts: $\varrho _{\alpha ,f}$ .

If A is a subset of $\alpha $ , we write $\varrho _{\alpha ,f}\restriction A=\langle \varrho _{\beta +1}(f(\beta +1)\mid \beta \in A\rangle $ , and so our recursive hypothesis tell us that $\varrho _{\alpha ,f}\in M_\alpha $ for every f, and this lets us define ${\mathbb {Q}}_{\alpha ,f}$ as the approximations of $\varrho _{\alpha ,f}$ . More rigorously, recall that we have defined the symmetric iteration ${\mathbb {P}}_\alpha $ , along with the direct limit of the generic semidirect products, $\mathcal G_\alpha $ . Then $\dot {\mathbb {Q}}_{\alpha ,f}$ is the forcing ordered by reverse inclusion on the interpretation of

$$\begin{align*}\left\{{\textstyle\int_{\vec\pi}}\dot\varrho_{\alpha,f}\restriction A\mathrel{}\middle|\mathrel{}\sup A<\alpha, {\textstyle\int_{\vec\pi}}\in\mathcal G_\alpha\right\}^\bullet.\end{align*}$$

For $E\subseteq \prod \operatorname {\mathrm {SC}}(\omega _\alpha )$ we may now define $\varrho _\alpha \restriction E=\langle \varrho _{\alpha ,f}\mid f\in E\rangle $ , and so if E is bounded in the product, i.e., there is $f\in \prod \operatorname {\mathrm {SC}}(\omega _\alpha )$ such that for all $g\in E$ and for all $\beta <\alpha $ , $g(\beta +1)<f(\beta +1)$ , and A is a bounded subset of $\alpha $ , our conditions are going to be approximations of $\varrho _\alpha $ , up to permutations of course, of the form $\langle \varrho _{\alpha ,f}\restriction A\mid f\in E\rangle $ , which we denote by $\varrho _\alpha \restriction (E,A)$ . And so ${\mathbb {Q}}_\alpha $ is given by the name

$$\begin{align*}\left\{{\textstyle\int_{\vec\pi}}\dot\varrho_\alpha\restriction(E,A)\mathrel{}\middle|\mathrel{} E,A\text{ are bounded and }{\textstyle\int_{\vec\pi}}\in\mathcal G_\alpha\right\}^\bullet.\end{align*}$$

We do not prove this statement here, but the idea is to simply utilise the upwards homogeneity of the previous steps and “correct” the coordinates one by one. One might ask how do we deal with the case where $\alpha>\omega $ , as a condition has seemingly infinitely many non-trivial coordinates. And the answer, as we repeatedly mention here, is utilising the previous limit cases where we condense this infinite amount of information, also in the form of ${\mathscr {G}}_\alpha $ being a subgroup of the full support product $\prod _{\beta <\alpha }{\mathscr {G}}_{\beta +1}$ .Footnote ²⁷ The complete proof can be found as Propositions 4.5 (for $\alpha =\omega $ ) and 4.15 in [Reference Karagila8]. The action of ${\mathscr {G}}_\alpha $ is coordinatewise, and it is important to stress at this point that we have this action where the initial segments of $\varrho _\alpha $ are “actual objects in $M_\alpha $ ”, so this is not applying automorphisms of a forcing, but rather applying permutations of each $\omega _{\beta +1}$ .

This again follows the same pattern as the successor steps, although here there is a notable complication in the case where $\alpha>\omega $ . We will outline the proof.

Now we only need to take care of the preservation of $V_{\omega +\alpha }^{M_\alpha }$ . Indeed, $M_\alpha [\varrho _\alpha ]$ contains the sequences $\varrho _{\beta +1}$ for $\beta <\alpha $ , all of which have rank smaller than $\omega +\alpha $ . Luckily, the symmetries of ${\mathbb {Q}}_\alpha $ will help us get rid of these unwelcomed sets.Footnote ²⁹

Combining these we have that if $\dot x$ is a symmetric name for a set of small rank, then all of its elements are decided by conditions with a uniform bound, meaning $\dot x$ is equal to an object in $M_\alpha $ . The proof of (1) is the standard homogeneity argument, and we have used it before in Theorem 2.2, so we will only outline the proof of (2).

The theorem is proved as Lemmas 4.10 and 4.27 in [Reference Karagila8]. This almost completes our decoding apparatus. We only need to worry about the $\varrho _{\alpha +1}$ now.

We define $R_\xi $ for $\xi <\omega _{\alpha +1}$ to be $V_{\omega +\alpha +1}^{M_\alpha [\varrho _{\alpha ,f^\alpha _\xi }]}$ . That is, we use $\varrho _{\alpha ,f^\alpha _\xi }$ as the “guide” for a new sequence in $V_{\omega +\alpha +1}$ . Those who kept track can guess now that $\varrho _{\alpha +1}$ is $\langle R_\xi \mid \xi <\omega _{\alpha +1}\rangle $ . We utilise the fact that the scale is permutable to ensure that any bounded part will be in $M_{\alpha +1}$ , as well as the rest of the permutability apparatus to ensure the upwards homogeneity. This is also the point where we see why the definition of ${\mathbb {Q}}_\alpha $ works in general, despite ${\mathbb {P}}_\alpha $ being a finite support iteration. The $\varrho _{\alpha ,f}$ are initial segments of those $\varrho _{\beta ,f^*}$ that come in the future, and even if $\operatorname {\mathrm {cf}}(\alpha )>\omega $ , we still end up with what we wanted to have.

With this we finish the discussion on the decoding apparatus. This is the main technical part of the construction. It is our sword and shield in our journey downwards. Now that we have that, we may venture deeper into the Hadean adventure that is the Bristol model.

4.1 Nature

In here we will define the Bristol model one step at a time by defining its von Neumann hierarchy in $L[c]$ . For this purpose it would be easier at times to use an increasing sequence modulo bounded sets, rather than the permutable families. Since the two are equivalent, we let $\{T^\alpha _\xi \mid \xi <\omega _{\alpha +1}\}$ denote a sequence obtained from $\mathcal A_\alpha $ .

We define the $\varrho _\alpha $ and $V_{\omega +\alpha }^M$ in tandem, and we will omit the M from the superscript where possible, as the definitions will be complicated enough. Let $\varrho _0=c$ and, as $V_\omega ^M$ is just $V_\omega ^L=L_\omega $ , it is defined. Suppose that for $\alpha $ , $\varrho _\alpha $ and $V_{\omega +\alpha }$ were defined.

If $\alpha $ is $0$ or a successor ordinal, define

(1) $$ \begin{align} V_{\omega+\alpha+1}=& \bigcup_{\xi<\omega_{\alpha+1}} V_{\omega+\alpha+1}^{L(V_{\omega+\alpha},\varrho_\alpha\restriction T^\alpha_\xi)}, \end{align} $$

(2) $$ \begin{align} \varrho_{\alpha+1}=& \left\langle V_{\omega+\alpha+1}^{L(V_{\omega+\alpha},\varrho_\alpha\restriction A^\alpha_\xi)}\mathrel{}\middle|\mathrel{}\xi<\omega_{\alpha+1}\right\rangle. \end{align} $$

If $\alpha $ is a limit ordinal, we define $V_\alpha =\bigcup _{\beta <\alpha } V_{\omega +\beta +1}$ , and we define ${\varrho _\alpha =\prod _{\beta <\alpha }\varrho _{\beta +1}}$ . And as before we write $\varrho _{\alpha ,f}$ to indicate the “thread” of the function f in this product. To define the next step we need an analogue of the $T^\alpha _\xi $ , we write $\overline {\varrho _{\alpha ,f}}$ to denote the sequence of $V_{\omega +\beta +1}^{L(V_\omega +\beta ,\varrho _\beta \restriction T^\beta _{f(\beta )})}$ for $\beta <\alpha $ .

And we now define

(3) $$ \begin{align} V_{\omega+\alpha+1}=& \bigcup_{\xi<\omega_{\alpha+1}} V_{\omega+\alpha+1}^{L(V_{\omega+\alpha},\overline{\varrho_{\alpha,f^\alpha_\xi}})}, \end{align} $$

(4) $$ \begin{align} \varrho_{\alpha+1}=& \left\langle V_{\omega+\alpha+1}^{L(V_{\omega+\alpha+1},\varrho_{\alpha,f^\alpha_\xi})}\mathrel{}\middle|\mathrel{}\xi<\omega_{\alpha+1}\right\rangle. \end{align} $$

Finally, $M=\bigcup _{\alpha \in \mathrm {Ord}} V_{\omega +\alpha +1}^M$ . The handwritten notes passed on to us by some of the members of the Bristol group indicate that the original line of thought was about $\mathbb R^M$ and $\mathcal {P}(\mathbb R)^M$ , rather than $V_{\omega +1}$ and $V_{\omega +2}$ . The arguments, moreover, as to why $V_{\omega +1}=V_{\omega +1}^{L(V_{\omega +1},\varrho _1\restriction T^\alpha _\xi )}$ , i.e., why no reals are added when defining $\mathcal {P}(\mathbb R)^M$ , was not written down in these notes. Instead it merely suggests “condensation”.

While restoring the original arguments is certainly beyond us, these would be equivalent, more or less, to the arguments we presented at the first step of Section 3.

4.2 Cause

In here we will define the Bristol model in tandem: define a symmetric extension, find it a generic in $L[c]$ , define a symmetric extension again, repeat ad ordinalum.

This definition was used in [Reference Karagila8], and you can very clearly see that this is the definition that we have in mind, as it very much dominates the approach given in Section 3. As such, we really have done all the work ahead of time.

We define $M_\alpha ,\varrho _\alpha $ as in the decoding apparatus, i.e., as the generic objects for the symmetric iteration. We now define

$$\begin{align*}M = \bigcup _{\alpha \in \mathrm {Ord}}M_\alpha =\bigcup _{\alpha \in \mathrm {Ord}}V_{\omega +\alpha }^{M_{\alpha +1}}.\end{align*}$$

4.3 Accident

In here we will first define a class-forcing, and then argue that we can just happen to find symmetrically generic filters in $L[c]$ .

Despite being the guiding view on the construction of the Bristol model, the actual argument for $M\models \mathsf {ZF}$ in [Reference Karagila8] is the one rising from this approach, as it is less “ad-hoc” and more structural and general. This is also the reason why the proof of the distributivity of successor steps in the original paper was proved directly, rather than the approach used in this article, which is more in line with “Cause”.

We first define the iteration. Let and ${\mathbb {Q}}_0=\operatorname {\mathrm {Add}}(\omega ,1)$ , ${\mathscr {G}}_0$ and ${\mathscr {F}}_0$ are the derived group and filter. Finally, $\dot \varrho _0$ is the canonical name for the Cohen real.

Suppose that ${\mathbb {P}}_\alpha $ is the finite support iteration of the symmetric extensions defined so far. In the case where $\alpha =\beta +1$ , we define $\dot {\mathbb {Q}}_\alpha $ as the name

$$\begin{align*}\left\{{\textstyle\int_{\vec\pi}}\left\langle\dot V_{\omega+\alpha}^{V[\varrho_\beta\restriction A^\beta_\xi]}\mathrel{}\middle|\mathrel{}\xi\in I\right\rangle^{\bullet}\mathrel{}\middle|\mathrel{}{\textstyle\int_{\vec\pi}}\in\mathcal G_\alpha, I\in[\omega_\alpha]^{<\omega_\alpha}\right\}^{\bullet}.\end{align*}$$

Where $\dot V_{\omega +\alpha }^{V[x]}$ denotes the name of the rank initial segment of the extension of $\mathsf {IS}_\beta $ by the set x. We then define $\dot \varrho _\alpha $ as the name for the generic of $\dot {\mathbb {Q}}_\alpha $ .

If $\alpha $ is a limit ordinal, we define ${\mathbb {P}}_\alpha $ as the finite support iteration. We next define $\dot {\mathbb {Q}}_\alpha $ in the same spirit. For $f\in \prod \operatorname {\mathrm {SC}}(\omega _{\alpha +1})$ we define

$$\begin{align*}\dot{\mathbb{Q}}_{\alpha,f}=\{{\textstyle\int_{\vec\pi}}\langle\dot\varrho_{\beta+1}(f(\beta+1))\mid\beta\in A\rangle\mid{\textstyle\int_{\vec\pi}}\in\mathcal G_\alpha, \sup A<\alpha\}^\bullet,\end{align*}$$

and we then define $\dot {\mathbb {Q}}_\alpha $ by adding the additional dimension of a bounded subset of $\prod \operatorname {\mathrm {SC}}(\omega _\alpha )$ . As before, ${\mathscr {G}}_\alpha $ and ${\mathscr {F}}_\alpha $ are the derived group and filter. Finally, $\dot \varrho _\alpha $ is the name of the generic filter. Moving on, the definition of $\dot {\mathbb {Q}}_{\alpha +1}$ and $\dot \varrho _{\alpha +1}$ is in line with what we have done so far.

The properties of the decoding apparatus imply the distributivity of each iterand. We need to be a bit more careful here, as we are not allowed to argue in $L[c]$ , instead we carry on a recursive hypothesis that for successor steps ${\mathbb {P}}_\alpha $ satisfies a chain condition that allows us to prove that $\dot {\mathbb {Q}}_\alpha $ will be isomorphic to $\operatorname {\mathrm {Add}}(\omega _\alpha ,1)^L$ .

The conditions of Theorem 2.8 are therefore satisfied. This implies that if G is any symmetrically L-generic for ${\mathbb {P}}$ , the class-length iteration, then $\mathsf {IS}^G\models \mathsf {ZF}$ , at the very least.

By recursion we can now show that each $\dot \varrho _\alpha $ has a symmetric interpretation inside $L[c]$ , and therefore we may find an interpretation of the model which is intermediate between L and $L[c]$ .

4.4 The basic properties of the Bristol model

We will see other ways of deducing Proposition 4.2 later on, in ways that will be significantly more informative and more general.

5.1 Corrections to Lemma 2.12/4.1

Lemma 2.12 is read in the context of the first steps. Namely, ${\mathbb {P}}$ is Cohen forcing, $\dot {\mathbb {Q}}$ is the name of the forcing adding $\varrho _1$ , denoted in that lemma by $\sigma $ . As such ${\mathsf {HS}}$ is the class of hereditarily symmetric names defined in the very first step.

In the article this lemma was used twice. The first consequence is the genericity of $\varrho _1$ ,Footnote ³² and the second is to show that $\dot {\mathbb {Q}}$ is $\sigma $ -distributive. The proof is very similar to the proof of Proposition 3.10. Nevertheless, when we have two permutations $\pi $ and $\tau $ , we want to move $\tau $ so that it agrees with $\pi $ .

For this we need not only that $\iota (\tau )\restriction A=\iota (\pi )\restriction A$ , but also that their inverses agree on A. In the case of the genericity of $\varrho _1$ we take $\pi =\operatorname {\mathrm {id}}$ , in which case this property holds trivially. Indeed, this is the very strategy of the proof of Proposition 3.10.

To prove that $\dot {\mathbb {Q}}$ is $\sigma $ -distributive we interpret the lemma above as saying that for a condition $p\in {\mathbb {P}}$ , there is an ordinal $\eta $ , such that deciding whether a condition from $\dot {\mathbb {Q}}$ , whose domain is at least $\eta $ , lies in $\dot D$ depends only on the permutation $\pi $ . We then utilise the fact ${\mathbb {P}}$ is c.c.c. to show that this $\eta $ can be bound uniformly depending on $\dot D$ . Now, if $\dot D_n$ are names for a sequence of dense open sets, we can uniformly bound all of them by some $\eta $ . Now if we take any condition whose domain is at least this $\eta $ , we may extend it into each $\dot D_n$ , but this means that such extension lies, in fact, in all the $\dot D_n$ simultaneously, and therefore the intersection is also dense.

Once we add the condition that $\iota (\pi )^{-1}\restriction A=\iota (\tau )^{-1}\restriction A$ , the proof as written in [Reference Karagila8] follows through. Alternatively, and perhaps more wisely, for proving the distributivity one should rely on the absoluteness proof that we used in this manuscript, which is also mentioned in the “Abyss”.

Lemma 4.1 is precisely the same, in the context of successor iterations. There is no need to repeat that which has been said.

5.2 Corrections to Proposition 4.4/4.15

These two propositions are dealing with limit iterations. They show that $\langle \varrho _\beta \mid \beta <\alpha \rangle $ is symmetrically generic for ${\mathbb {P}}_\alpha $ , with Proposition 4.4 dealing with the case $\alpha =\omega $ , which was treated separately in the article.

In the article the proof takes a symmetrically dense open set D, and an excellent support $\vec H$ witnessing that. We then generate some sequence of domains such that when we extend the condition $\langle \varrho _\beta \restriction X_\beta \mid \beta <\alpha \rangle $ into D, then we can “correct” it using automorphisms which lie in $\vec H$ . Namely, each coordinate of $\vec H$ is of the form $\operatorname {\mathrm {fix}}(\mathcal B_\beta )$ , and we take $X_\beta $ to be $\sup \operatorname {\mathrm {dom}}\mathcal B_\beta +1$ , and therefore our starting condition is $\langle \varrho _\beta \restriction X_\beta \mid \beta <\alpha \rangle $ .

The idea is fine, and it is somehow intuitively clear what should happen. However the proof described above, taken from the “Abyss”, will not work. The first hint is obvious: $\varrho _\beta $ has domain $\omega _\beta $ , and $X_\beta $ is a bounded subset of $\omega _{\beta +1}$ . We can resolve this issue by considering $\varrho _{\beta +1}\restriction X_\beta $ , but this raises the obvious question, what should we do with limit steps and with $\varrho _0$ ?

Let us prove this proposition in its general form.

Indeed, the main point is the fact that we may assume that the first coordinate, for which true genericity is already assumed, is compatible with the Cohen real, and then we can modify any further coordinates recursively using upward homogeneity, combined with the fact that we only need to change things outside of the domains we found, which means that the needed automorphisms can be found in $\vec H$ .

6.1 Limitations

Despite this handsome accommodation of large cardinal assumptions in the ground model, as well as in the Bristol model itself, we can put a stop to this. Indeed, if $\kappa $ is a supercompact cardinal, then there is a notable shortage of $\square ^*_\lambda $ sequences for singular $\lambda $ such that $\operatorname {\mathrm {cf}}(\lambda )<\kappa <\lambda $ . We may ask ourselves, perhaps we can salvage permutable scales without having $\square ^*_\lambda $ ?

Recall that we are still working under the assumption of $\mathsf {GCH}$ . Removing it will require adding $2^\lambda <2^{\lambda ^+}$ , which itself is a harmless assumption as it holds for all strong limit cardinals above $\kappa $ .

Proof Let $F=\{f_\alpha \mid \alpha <\lambda ^+\}$ be a scale in $\prod \operatorname {\mathrm {SC}}(\lambda )$ , and let $\pi $ be a permutation of $\lambda ^+$ , not necessarily bounded, which is not implemented by any sequence of permutations, $\vec \pi $ . Pick $j\colon V\to W$ an elementary embedding with critical point $\kappa $ witnessing that $\kappa $ is at least $\lambda ^+$ -supercompact, so in particular $j(\kappa )>\lambda ^+$ . We denote by $G=\{g_\alpha \mid \alpha <j(\lambda ^+)\}$ the scale $j(F)$ , which is a scale in $j(\prod \operatorname {\mathrm {SC}}(\lambda ))$ which is equal to $\prod \operatorname {\mathrm {SC}}(j(\lambda ))^M$ .

Let $\nu =\sup j"\lambda ^+$ , then $\nu $ is closed under $j(\pi )$ , so we can let $\tau $ denote $j(\pi )\restriction \nu $ , which is a bounded permutation of $j(\lambda ^+)$ .

Suppose that there was a sequence of permutations $\vec \tau $ that implemented $\tau $ , then for $\alpha <\lambda ^+$ and $\mu <\lambda $ , we have $g_{j(\alpha )}(j(\mu ^+))=j(f_\alpha (\mu ^+))$ and $\tau (j(\alpha ))=j(\pi (\alpha ))$ . Combining this with the assumption that $\vec \tau $ implements $\tau $ , we get that

$$\begin{align*}\tau_{j(\mu^+)}(j(f_\alpha(\mu^+)))=j(f_{\pi(\alpha)}(\mu^+)).\end{align*}$$

We use this to define $\pi _{\mu ^+}\colon \mu ^+\to \mu ^+$ . If $\tau _{j(\mu ^+)}(j(\xi ))=j(\zeta )$ for some $\zeta $ , define $\pi _{\mu ^+}(\xi )=\zeta $ ; otherwise $\pi _{\mu ^+}(\xi )=\xi $ . This is well-defined, since $\tau _{j(\mu ^+)}$ is itself a permutation of $j(\mu ^+)$ , and so $j(\zeta )<j(\mu ^+)$ . This is also a permutation by similar reasons.

Finally, we claim that $\vec \pi $ implements $\pi $ . Fix $\alpha <\lambda ^+$ , then for any large enough $\mu ^+<\lambda $ , $\tau _{j(\mu ^+)}(j(f_\alpha (\mu ^+)))=j(f_{\pi (\alpha )}(\mu ^+))$ , which means that we defined $\pi _{\mu ^+}$ to be exactly $f_{\pi (\alpha )}(\mu ^+)$ .

We have shown that if F can be used to implement all bounded permutations, then it can be used to implement all permutations, but that is definitely impossible on grounds of cardinal arithmetic.